Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem

(1)

R E S E A R C H

Open Access

Algorithms with strong convergence for the

split common solution of the feasibility

problem and ﬁxed point problem

Yonghong Yao

1

_{, Ravi P Agarwal}

2,3

_{, Mihai Postolache}

4*

_{and Yeong-Cheng Liou}

5 *_{Correspondence:}

[email protected] 4_{Faculty of Applied Sciences,}

University ‘Politehnica’ of Bucharest, Splaiul Independentei 313, Bucharest, 060042, Romania Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to study the split feasibility problem and ﬁxed point problem involved in the pseudocontractive mappings. We construct an iterative algorithm and prove its strong convergence.

MSC: 47J25; 47H09; 65J15; 90C25

Keywords: split feasibility problem; ﬁxed point problem; pseudocontractive mapping

1 Introduction

LetHandHbe two real Hilbert spaces and letC⊂HandQ⊂H be two nonempty, closed, and convex sets. LetA:H→Hbe a bounded linear operator with its adjointA∗.

LetS:H→HandT:H→Hbe two nonlinear mappings.

The purpose of this paper is to study the following split feasibility problem and ﬁxed point problem:

Findx∗∈C∩Fix(T) such thatAx∗∈Q∩Fix(S). (.) Special cases:

(i) Finding a pointx∗which satisﬁes

x∗∈C and Ax∗∈Q. (.)

This problem, referred to as the split feasibility problem, was introduced by Censor and Elfving [], modeling phase retrieval and other image restoration problems, and further studied by many researchers; see, for instance, [–].

(ii) Find a pointx∗with the property

x∗∈Fix(T) and Ax∗∈Fix(S). (.)

This problem, referred to as the split common ﬁxed point problem, was ﬁrst introduced by Censor and Segal [].

Next, we recall some existing algorithms for solving (.)-(.) in the literature.

(2)

In order to solve (.), Censor and Elfving [] introduced the following algorithm:

xn+=A–PQ

PA(C)(Axn)

, n∈N, (.)

whereCandQare closed and convex sets inRn_,_A_{is a full rank}_n_×_n_{matrix and}_A₍_C_{) =}

{y∈Rn_|_y₌_Ax_,_x_∈_C_}.

Now (.) is not popular because it involves the computation of the inverseA–_.

A more popular algorithm that solves (.) seems to be theCQalgorithm presented by Byrne [, ]:

xn+=PCxn–τA∗(I–PQ)Axn, n∈N, (.)

whereτ∈(,_L), withLbeing the largest eigenvalue of the matrixA∗A. Note thatx∗solves (.) if and only ifx∗solves the ﬁxed point equation

x∗=PC

I–λA∗(I–PQ)A

x∗. (.)

The above equivalence relation (.) reminds us to use ﬁxed point method to solve (.). Many authors have given a continuation of the study on the CQ algorithm and its variant form. For related work, please refer to [–]. Especially, the following regularized method was presented by Xu []:

xn+=PC( –αnγn)xn–γnA∗(I–PQ)Axn

, n∈N. (.)

It should be pointed out that (.) can be used to ﬁnd the minimum norm solution of (.). For solving (.), Censor and Segal [] invented an algorithm which generates a sequence {xn}according to the iterative procedure:

xn+=Txn–γA∗(I–S)Axn, n∈N. (.)

Note that (.) is more general than (.). Some further generations of this algorithm were studied by Moudaﬁ [] and Wang and Xu [] and others; see, for example, [–].

Motivated by the results in this direction, the purpose of this paper is to study the split feasibility problem and the ﬁxed point problem involved in the pseudocontractive map-pings. We construct an iterative algorithm and prove its strong convergence.

2 Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · , respectively. LetC

be a nonempty, closed, and convex subset ofH.

Recall that a mappingT:C→Cis calledpseudocontractiveif

Tx–Ty,x–y ≤ x–y 

for allx,y∈C. It is well known thatTis pseudocontractive if and only if

(3)

for allx,y∈C. A mappingT:C→Cis calledL-Lipschitzianif there existsL>  such that

Tx–Ty ≤L x–y

for allx,y∈C. IfL= , we callTnonexpansive.

We will useFix(T) to denote the set of ﬁxed points ofT, that is,

Fix(T) ={x∈C:x=Tx}.

We know that the metric projectionPC:H→Csatisﬁes

x–PC(x)=inf

x–y :y∈C.

It is well known that the metric projectionPC:H→Cis ﬁrmly nonexpansive, that is,

x–y,PC(x) –PC(y)≥PC(x) –PC(y)

⇔ PC(x) –PC(y)≤ x–y –(I–PC)x– (I–PC)y (.)

for allx,y∈H.

For allx,y∈H, the following conclusions hold:

tx+ ( –t)y=t x + ( –t) y –t( –t) x–y , t∈[, ], (.)

x+y = x + x,y+ y , (.)

and

x+y ≤ x + y,x+y. (.)

Lemma .([]) Let H be a real Hilbert space,C a closed convex subset of H.Let T:C→ C be a continuous pseudocontractive mapping.Then

(i) Fix(T)is a closed convex subset ofC,

(ii) (I–T)is demiclosed at zero.

Lemma .([]) Assume that{an}is a sequence of nonnegative real numbers such that

an+≤( –γn)an+δn, n∈N,

where{γn}is a sequence in(, )and{δn}is a sequence such that

() ∞_n=γn=∞;

() lim sup_n_→∞δn γn≤or

∞

n=|δn|<∞. Thenlimn→∞an= .

Lemma .([]) Let{wn}be a sequence of real numbers.Assume{wn}does not decrease at inﬁnity,that is,there exists at least a subsequence{wnk}of{wn}such that wnk≤wnk+ for all k≥.For every n≥N,deﬁne an integer sequence{τ(n)}as

(4)

Thenτ(n)→ ∞as n→ ∞and for all n≥N

max{wτ(n),wn} ≤wτ(n)+.

3 Main results

LetHandHbe two real Hilbert spaces and letC⊂HandQ⊂H be two nonempty closed convex sets. LetA:H→Hbe a bounded linear operator with its adjointA∗. Let

S:H→Hnonexpansive mapping and letT:H→Hbe anL-Lipschitzian pseudocon-tractive mapping withL> .

We useto denote the set of solutions of (.), that is,

=x∗|x∗∈C∩Fix(T),Ax∗∈Q∩Fix(S). In the sequel, we assume=∅.

Now, we present our algorithm for ﬁndingx∗∈.

Algorithm . For ﬁxedu∈Handx∈Harbitrarily, let{xn}be a sequence deﬁned by

⎧ ⎨ ⎩

un=PC[αnu+ ( –αn)(xn–δA∗(I–SPQ)Axn)],

xn+= ( –βn)un+βnT(( –γn)un+γnTun), n∈N,

(.)

where{αn}n∈N,{βn}n∈N, and{γn}n∈Nare three real number sequences in (, ) andδis a constant in (, _A).

Theorem . Assume the following conditions are satisﬁed:

(C) limn→∞αn= ;

(C) ∞_n=αn=∞;

(C)  <a<βn<c<γn<b<√  +L_+.

Then the sequence{xn}generated by algorithm(.)converges strongly to the point x∗,

given by x∗=P(u).

Proof Setx∗=P(u). Then we havex∗∈C∩Fix(T) andAx∗∈Q∩Fix(S). Setzn=PQAxn

andyn=αnu+ ( –αn)(xn–δA∗(I–SPQ)Axn) for alln∈N. Thusun=PCynfor alln∈N.

SincePCandPQare nonexpansive, we have

zn–Ax∗=PQAxn–PQAx∗≤Axn–Ax∗ (.)

and

un–x∗=PCyn–PCx∗≤yn–x∗. (.)

By (.), we get

Szn–Ax∗=SPQAxn–SPQAx∗

≤PQAxn–PQAx∗

(5)

From (.), we have

Tun–x∗≤un–x∗+ Tun–un  (.)

and

T( –γn)un+γnTun

–x∗

≤( –γn)

un–x∗+γn

Tun–x∗

+( –γn)un+γnTun–T

( –γn)un+γnTun. (.)

Applying equality (.), we have

( –γn)un+γnTun–T

( –γn)un+γnTun 

=( –γn)

un–T( –γn)un+γnTun

+γn

Tun–T( –γn)un+γnTun

= ( –γn)un–T

( –γn)un+γnTun+γnTun–T

( –γn)un+γnTun

–γn( –γn) un–Tun . (.)

SinceTisL-Lipschitzian andun– (( –γn)un+γnTun) =γn(un–Tun), by (.), we get

( –γn)un+γnTun–T

≤( –γn)un–T

( –γn)un+γnTun+γnL un–Tun 

–γn( –γn) un–Tun 

= ( –γn)un–T

+γ_nL+γ_n–γn

un–Tun . (.)

By (.) and (.), we have

( –γn)

un–x∗+γn

Tun–x∗

= ( –γn)un–x∗+γnTun–x∗–γn( –γn) un–Tun 

≤( –γn)un–x∗+γnun–x∗+ un–Tun 

–γn( –γn) un–Tun 

=un–x∗+γ_n un–Tun . (.)

From (.), (.), and (.), we deduce T( –γn)un+γnTun

–x∗

≤un–x∗ 

+ ( –γn)un–T

–γn

 – γn–γnL

un–Tun . (.)

Sinceγn<b<√_+L_+, we derive that

(6)

This together with (.) implies that

_T

( –γn)un+γnTun

–x∗

≤un–x∗ 

+ ( –γn)un–T

. (.)

By (.), (.), (.), and (C), we have

xn+–x∗=( –βn)un+βnT

–x∗

= ( –βn)un–x∗ 

+βnT

–x∗

–βn( –βn)un–T

≤un–x∗ 

–βn(γn–βn)T

–x∗

≤un–x∗ 

. (.)

By the convexity of the norm and by using (.), we get

yn–x∗=αn

u–x∗+ ( –αn)

xn–x∗+δA∗(Szn–Axn) ≤( –αn)xn–x∗+δA∗(Szn–Axn)



+αnu–x∗ 

= ( –αn)xn–x∗+δA∗(Szn–Axn) 

+ δxn–x∗,A∗(Szn–Axn)

+αnu–x∗ 

. (.)

SinceAis a linear operator with its adjointA∗, we have

xn–x∗,A∗(Szn–Axn) =Axn–x∗,Szn–Axn

=Axn–Ax∗+Szn–Axn– (Szn–Axn),Szn–Axn

=Szn–Ax∗,Szn–Axn– Szn–Axn . (.) Again using (.), we obtain

Szn–Ax∗,Szn–Axn=

Szn–Ax

∗

+ Szn–Axn –Axn–Ax∗. (.) From (.), (.), and (.), we get

xn–x∗,A∗(Szn–Axn)=

Szn–Ax

∗

+ Szn–Axn –Axn–Ax∗

– Szn–Axn 

≤

Axn–Ax

∗

– zn–Axn + Szn–Axn 

–Axn–Ax∗ 

– Szn–Axn 

= –

 zn–Axn

_–

 Szn–Axn

(7)

Substituting (.) into (.) we deduce

yn–x∗≤( –αn)

xn–x∗+δ A  Szn–Axn 

+ δ

–

 zn–Axn

_–

 Szn–Axn



+αnu–x∗ 

= ( –αn)xn–x∗+

δ A –δ Szn–Axn 

–δ zn–Axn +αnu–x∗ 

≤( –αn)xn–x∗ 

+αnu–x∗ 

. (.)

From (.), (.), and (.), we get xn+–x∗



≤yn–x∗

≤( –αn)xn–x∗ 

+αnu–x∗ 

≤maxxn–x∗ 

,u–x∗. The boundedness of the sequence{xn}yields our result.

Using the ﬁrmly nonexpansivenessity ofPC(.), we have

un–x∗=PCyn–x∗≤yn–x∗– PCyn–yn 

=yn–x∗ 

– un–yn . (.)

Thus

xn+–x∗≤un–x∗

≤yn–x∗– un–yn 

≤( –αn)xn–x∗ 

+αnu–x∗ 

– un–yn .

It follows that

un–yn ≤xn–x∗–xn+–x∗+αnu–x∗. (.)

Next, we consider two possible cases.

Case . Assume there exists some integerm>  such that{ xn–x∗ }is decreasing for alln≥m. In this case, we know thatlimn→∞ xn–x∗ exists. From (.), we deduce

lim

n→∞ un–yn = . (.)

Returning to (.), we have xn+–x∗≤yn–x∗

≤( –αn)xn–x∗ 

+ ( –αn)

δ A –δ Szn–Axn 

– ( –αn)δ zn–Axn +αnu–x∗ 

(8)

Hence,

( –αn)

δ–δ A  Szn–Axn + ( –αn)δ zn–Axn 

≤xn–x∗–xn+–x∗+αnu–x∗ 

,

which implies that

lim

n→∞ Szn–Axn =nlim→∞ zn–Axn = . (.)

So,

lim

n→∞ Szn–zn = . (.)

Note that

yn–xn =δA∗(SPQ–I)Axn+αn

xn–δA∗(I–SPQ)Axn–u

≤δ A Szn–Axn +αnxn–δA∗(I–SPQ)Axn–u.

It follows from (.) that

lim

n→∞ xn–yn = . (.)

From (.), (.), and (.), we deduce

xn+–x∗≤un–x∗–βn(γn–βn)un–T

( –γn)un+γnTun

≤xn–x∗+αnu–x∗

–βn(γn–βn)un–T

( –γn)un+γnTun.

It follows that

βn(γn–βn)un–T

≤xn–x∗–xn+–x∗+αnu–x∗.

Therefore,

lim n→∞un–T

( –γn)un+γnTun= . (.)

Observe that

un–Tun ≤un–T( –γn)un+γnTun+T

–Tun

≤un–T( –γn)un+γnTun+Lγn un–Tun .

Thus,

un–Tun ≤ 

 –Lγn

(9)

This together with (.) implies that

lim

n→∞ un–Tun = . (.)

Now, we show that

lim sup n→∞

u–x∗,yn–x∗

≤.

Choose a subsequence{yni}of{yn}such that

lim sup n→∞

u–x∗,yn–x∗

=lim

i→∞

u–x∗,yni–x∗

. (.)

Since the sequence{yni}is bounded, we can choose a subsequence{yn_ij}of{yni}such that

yn_ij z. For the sake of convenience, we assume (without loss of generality) thatyni z.

Consequently, we derive from the above conclusions that

xni z, uni z, Axni Az and zni Az. (.)

Applying Lemma ., we deduce

z∈Fix(T) and Az∈Fix(S).

Note thatuni=PCyni∈Candzni=PQAxni∈Q. From (.), we deduce

z∈C and Az∈Q.

To this end, we deduce

z∈C∩Fix(T) and Az∈Q∩Fix(S).

That is to say,z∈. Therefore,

lim sup n→∞

u–x∗,yn–x∗=lim i→∞

u–x∗,yni–x∗

=lim i→∞

u–x∗,z–x∗

≤. (.)

Using (.), we have

xn+–x∗≤yn–x∗

=( –αn)

xn–δA∗(I–SPQ)Axn–x∗

+αn

u–x∗

≤( –αn)xn–δA∗(I–SPQ)Axn–x∗ 

+ αn

u–x∗,yn–x∗

≤( –αn)xn–x∗ 

+ αn

u–x∗,yn–x∗

. (.)

(10)

Case . Assume there exists an integernsuch that

xn–x∗≤xn+–x∗.

Setωn={ xn–x∗ }. Then we have

ωn≤ωn+.

Deﬁne an integer sequence{τn}for alln≥nas follows:

τ(n) =max{l∈N|n≤l≤n,ωl≤ωl+}.

It is clear thatτ(n) is a non-decreasing sequence satisfying

lim

n→∞τ(n) =∞

and

ωτ(n)≤ωτ(n)+,

for alln≥n.

By a similar argument to that of Case , we can obtain

lim

n→∞ uτ(n)–yτ(n) =nlim→∞ xτ(n)–yτ(n) = , lim

n→∞ Szτ(n)–Axτ(n) =nlim→∞ zτ(n)–Axτ(n) =nlim→∞ Szτ(n)–zτ(n) = ,

and

lim

n→∞ uτ(n)–Tuτ(n) = .

This implies that

ωw(yτ(n))⊂.

Thus, we obtain

lim sup n→∞

u–x∗,yτ(n)–x∗≤. (.)

Sinceωτ(n)≤ωτ(n)+, we have from (.) that

ω_τ_(n)≤ω_τ_(n)+≤( –ατ(n))ωτ(n)+ ατ(n)

u–x∗,yτ(n)–x∗. (.) It follows that

ω_τ_(n)≤u–x∗,yτ(n)–x∗

(11)

Combining (.) and (.), we have

lim sup

n→∞ ωτ(n)≤,

and hence

lim

n→∞ωτ(n)= . (.)

By (.), we obtain

lim sup n→∞

ω_τ(n)+≤lim sup n→∞

ω_τ(n).

This together with (.) implies that

lim

n→∞ωτ(n)+= .

Applying Lemma . to get

≤ωn≤max{ωτ(n),ωτ(n)+}.

Therefore,ωn→. That is,xn→x∗. This completes the proof.

Algorithm . Forx∈Harbitrarily, let{xn}be a sequence deﬁned by

⎧ ⎨ ⎩

un=PC[( –αn)(xn–δA∗(I–SPQ)Axn)],

(.)

where{αn}n∈N,{βn}n∈N, and{γn}n∈Nare three real number sequences in (, ) andδis a

constant in (, _A).

Corollary . Assume the following conditions are satisﬁed:

(C) lim_n_→∞αn= ;

(C) ∞_n=αn=∞;

(C)  <a<βn<c<γn<b<√  +L_+.

Then the sequence{xn}generated by algorithm(.)converges strongly to x∗=P(),

which is the minimum norm in.

Algorithm . For ﬁxedu∈Handx∈Harbitrarily, let{xn}be a sequence deﬁned by

xn+=PCαnu+ ( –αn)

xn–δA∗(I–PQ)Axn, n∈N, (.)

where{αn}n∈Nis a real number sequence in (, ) andδis a constant in (, _A).

(12)

(C) ∞_n=αn=∞.

Then the sequence{xn}generated by algorithm(.)converges strongly to x∗=P(u). Algorithm . Forx∈Harbitrarily, let{xn}be a sequence deﬁned by

xn+=PC( –αn)

xn–δA∗(I–PQ)Axn, n∈N, (.) where{αn}n∈Nis a real number sequence in (, ) andδis a constant in (, _A).

Corollary . Suppose,the set of the solutions of (.),is nonempty.Assume the follow-ing conditions are satisﬁed:

(C) ∞_n=αn=∞.

Then the sequence{xn}generated by algorithm(.)converges strongly to x∗=P(), which is the minimum norm in.

Algorithm . For ﬁxedu∈Handx∈Harbitrarily, let{xn}be a sequence deﬁned by

⎧ ⎨ ⎩

un=αnu+ ( –αn)(xn–δA∗(I–S)Axn),

(.)

where{αn}n∈N,{βn}n∈N, and{γn}n∈Nare three real number sequences in (, ) andδis a

constant in (,  A).

Corollary . Suppose,the set of the solutions of (.),is nonempty.Assume the fol-lowing conditions are satisﬁed:

(C) ∞_n=αn=∞;

(C)  <a<βn<c<γn<b<√_+L_+.

Then the sequence{xn}generated by algorithm(.)converges strongly to x∗=P(u).

Algorithm . For andx∈Harbitrarily, let{xn}be a sequence deﬁned by ⎧

⎨ ⎩

un= ( –αn)(xn–δA∗(I–S)Axn),

(.)

where{αn}n∈N,{βn}n∈N, and{γn}n∈Nare three real number sequences in (, ) andδis a constant in (, _A).

Corollary . Suppose,the set of the solutions of (.),is nonempty.Assume the fol-lowing conditions are satisﬁed:

(C) ∞_n=αn=∞;

(C)  <a<βn<c<γn<b<√_+L_+.

(13)

Example . LetH=H=Rwith the inner product deﬁned byx,y=xyfor allx,y∈R

and the standard norm| · |. LetC= [,∞) andQ=R. LetSx=x_–  for allx∈Qand let

Tx=x–  +_x+ for allx∈C. LetAx= –x_ for allx∈R. ThenAis a bounded linear operator with its adjointA∗=A. Observe thatFix(T) =  andFix(S) = –. It is easy to see that

Tx–Ty,x–y=

x–  + 

x+ –y+  – 

y+ ,x–y

≤

 – 

(x+ )(y+ )

|x–y|

≤ |x–y|,

and

|Tx–Ty| ≤x–  + 

x+ –y+  – 

y+ 

≤ – 

(x+ )(y+ ) |x–y|

≤|x–y|,

for allx,y∈C. But T  

–T()=  >

 .

Hence,T is a Lipschitzian pseudocontractive mapping but not a nonexpansive one. Note that A = A∗ =_. Letu=  andδ=_. Then we have

un=PC

αn+ ( –αn)

xn–

  ×

–I

 × I +  × –xn  =PC

αn+ ( –αn)

 xn+

 

.

Letβn=_ andγn=_ for alln. It is not hard to compute that

|xn+– | ≤ |un– |

≤( –αn)

_xn+

– 

≤ |xn– | ≤ · · · ≤   n

|x– |,

(14)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China.}2_{Department of Mathematics, Texas}

A&M University, Kingsville, USA.3_{Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz} University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.4_{Faculty of Applied Sciences, University ‘Politehnica’ of Bucharest,} Splaiul Independentei 313, Bucharest, 060042, Romania.5_{Department of Information Management, Cheng Shiu} University, Kaohsiung, 833, Taiwan.

Acknowledgements

YY was supported in part by NSFC 71161001-G0105. Y-CL was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.

Received: 11 April 2014 Accepted: 25 August 2014 Published:02 Sep 2014

References

1. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms8, 221-239 (1994)

2. Censor, Y, Bortfeld, T, Martin, B, Troﬁmov, A: A uniﬁed approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol.51, 2353-2365 (2006)

3. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl.21, 2071-2084 (2005)

4. Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl.327, 1244-1256 (2007)

5. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl.18, 441-453 (2002)

6. Xu, HK: Iterative methods for the split feasibility problem in inﬁnite-dimensional Hilbert spaces. Inverse Probl.26, 105018 (2010)

7. Byrne, C: A uniﬁed treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl.20, 103-120 (2004)

8. Censor, Y, Segal, A: The split common ﬁxed point problem for directed operators. J. Convex Anal.16, 587-600 (2009) 9. Yu, X, Shahzad, N, Yao, Y: Implicit and explicit algorithms for solving the split feasibility problem. Optim. Lett. (2012).

doi:10.1007/s11590-011-0340-0

10. Qu, B, Xiu, N: A note on the CQ algorithm for the split feasibility problem. Inverse Probl.21, 1655-1665 (2005) 11. Zhao, J, Yang, Q: Several solution methods for the split feasibility problem. Inverse Probl.21, 1791-1799 (2005) 12. Dang, Y, Gao, Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl.27,

015007 (2011)

13. Wang, F, Xu, HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. (2010). doi:10.1155/2010/102085

14. Wang, Z, Yang, Q, Yang, Y: The relaxed inexact projection methods for the split feasibility problem. Appl. Math. Comput. (2010). doi:10.1016/j.amc.2010.11.058

15. Yao, Y, Wu, J, Liou, YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal.2012, Article ID 140679 (2012)

16. Yao, Y, Postolache, M, Liou, YC: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl.2013, Article ID 201 (2013)

17. Moudafi, A: The split common fixed-point problem for demicontractive mappings. Inverse Probl.26, 055007 (2010) 18. Wang, F, Xu, HK: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal.74, 4105-4111 (2011) 19. Zhao, J, He, S: Alternating Mann iterative algorithms for the split common fixed-point problem of

quasi-nonexpansive mappings. Fixed Point Theory Appl.2013, Article ID 288 (2013)

20. Zhao, J, He, S: Simultaneous iterative algorithms for the split common ﬁxed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms. Fixed Point Theory Appl.

2014, Article ID 73 (2014)

21. Chang, SS, Kim, J, Cho, YJ, Sim, J: Weak and strong convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces. Fixed Point Theory Appl.2014, Article ID 11 (2014)

22. Cui, H, Wang, F: Iterative methods for the split common ﬁxed point problem in Hilbert spaces. Fixed Point Theory Appl.2014, Article ID 78 (2014)

23. Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudocontractions in Banach spaces. Nonlinear Anal.70, 4039-4046 (2009)

24. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.66, 240-256 (2002)

25. Mainge, PE: Approximation methods for common ﬁxed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl.325, 469-479 (2007)

10.1186/1687-1812-2014-183